Quote by joe_cool2
The equation for force I must use, then, is:
F = ma + v(dm/dt)

Why is that the equation you "must" use?
There are three choices for addressing variable mass systems in the context of Newton's laws:
 [itex]\vec F=m\vec a[/itex], in which case thrust is an external force,
 [itex]\vec F=\frac{d\vec p}{dt}[/itex], in which case force is not frame invariant, or
 Newton's laws are valid only for constant mass particles, in which case Newton's laws do not apply (at least not directly) to variable mass systems.
Your definition of force, [itex]\vec F = m\vec a +\frac{dm}{dt}\vec v[/itex] corresponds to choice #2. This yields a force that is neither frame invariant nor equal to thrust. Most people who work with rockets use option #1 or #3.